{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The following experimenting is based on https://github.com/pygae/clifford/blob/master/docs/ConformalGeometricAlgebra.ipynb ( see it at https://nbviewer.jupyter.org/github/pygae/clifford/blob/master/docs/ConformalGeometricAlgebra.ipynb )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Requirement already satisfied: clifford in /srv/conda/lib/python3.6/site-packages\n",
      "Requirement already satisfied: future in /srv/conda/lib/python3.6/site-packages (from clifford)\n",
      "Requirement already satisfied: scipy in /srv/conda/lib/python3.6/site-packages (from clifford)\n",
      "Requirement already satisfied: numpy in /srv/conda/lib/python3.6/site-packages (from clifford)\n",
      "Requirement already satisfied: numba in /srv/conda/lib/python3.6/site-packages (from clifford)\n",
      "Requirement already satisfied: llvmlite>=0.24.0dev0 in /srv/conda/lib/python3.6/site-packages (from numba->clifford)\n",
      "\u001b[33mYou are using pip version 9.0.3, however version 18.0 is available.\n",
      "You should consider upgrading via the 'pip install --upgrade pip' command.\u001b[0m\n",
      "Requirement already satisfied: matplotlib in /srv/conda/lib/python3.6/site-packages\n",
      "Requirement already satisfied: python-dateutil>=2.1 in /srv/conda/lib/python3.6/site-packages (from matplotlib)\n",
      "Requirement already satisfied: numpy>=1.7.1 in /srv/conda/lib/python3.6/site-packages (from matplotlib)\n",
      "Requirement already satisfied: six>=1.10 in /srv/conda/lib/python3.6/site-packages (from matplotlib)\n",
      "Requirement already satisfied: cycler>=0.10 in /srv/conda/lib/python3.6/site-packages (from matplotlib)\n",
      "Requirement already satisfied: pytz in /srv/conda/lib/python3.6/site-packages (from matplotlib)\n",
      "Requirement already satisfied: pyparsing!=2.0.4,!=2.1.2,!=2.1.6,>=2.0.1 in /srv/conda/lib/python3.6/site-packages (from matplotlib)\n",
      "Requirement already satisfied: kiwisolver>=1.0.1 in /srv/conda/lib/python3.6/site-packages (from matplotlib)\n",
      "Requirement already satisfied: setuptools in /srv/conda/lib/python3.6/site-packages (from kiwisolver>=1.0.1->matplotlib)\n",
      "\u001b[33mYou are using pip version 9.0.3, however version 18.0 is available.\n",
      "You should consider upgrading via the 'pip install --upgrade pip' command.\u001b[0m\n"
     ]
    }
   ],
   "source": [
    "!pip install clifford\n",
    "!pip install matplotlib"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "from IPython.display import display, Math, Latex, Markdown"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 133,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "%config InlineBackend.figure_format = 'svg'\n",
    "import matplotlib.pyplot as plt\n",
    "from mpl_toolkits.mplot3d import Axes3D\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 120,
   "metadata": {},
   "outputs": [],
   "source": [
    "def str_ga(*gas, expand_wedge=False):\n",
    "    rets = []\n",
    "    for ga in gas:\n",
    "        ret = []\n",
    "        first = True\n",
    "        for i, v in enumerate(ga.value):\n",
    "            if not first:\n",
    "                if v > 0:\n",
    "                    ret.append(' + ')\n",
    "            if v < 0:\n",
    "                ret.append(' - ')\n",
    "            if v != 0:\n",
    "                v = abs(v)\n",
    "                blade_tuple = ga.layout.bladeTupList[i]\n",
    "                if blade_tuple != ():\n",
    "                    if v != 1:\n",
    "                        ret.append('%g ' % v)\n",
    "                    if expand_wedge:\n",
    "                        ret.append(' \\\\wedge '.join([ ('e_%d' % base) for base in blade_tuple ]))\n",
    "                    else:\n",
    "                        ret.append('e_{%s}' % ''.join([ ('%d' % base) for base in blade_tuple ]))\n",
    "                else:\n",
    "                    ret.append('%g ' % v)\n",
    "                first = False\n",
    "        if ret == []:\n",
    "            ret.append('0')\n",
    "        rets.append('%s' % ''.join(ret))\n",
    "    return rets"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 108,
   "metadata": {},
   "outputs": [],
   "source": [
    "def print_ga(*gas, expand_wedge=False):\n",
    "    for ga_latex in str_ga(*gas, expand_wedge=False):\n",
    "        display(Math(ga_latex))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Conformal Geometric Algebra (CGA) is a projective geometry tool which allows conformal transformations to be implemented with rotations.   To do this,  the original geometric algebra is extended by two dimensions, one of positive signature $e_+$ and one of negative signature $e_-$. Thus, if we started with $G_p$,  the conformal algebra is  $G_{p+1,1}$.\n",
    "\n",
    "It is convenient to define a *null* basis given by \n",
    "\n",
    "$$e_{o} = \\frac{1}{2}(e_{-} -e_{+})\\\\e_{\\infty} = e_{-}+e_{+}$$\n",
    "\n",
    "A vector in the original space $x$ is *up-projected* into a conformal vector $X$  by \n",
    "\n",
    "$$X = x + \\frac{1}{2} x^2 e_{\\infty} +e_o $$\n",
    "\n",
    "\n",
    "To map a conformal vector back into a vector from the original space, the vector is first normalized, then rejected from the minkowski plane $E_0$,\n",
    "\n",
    "\n",
    "$$ X = \\frac{X}{X \\cdot e_{\\infty}}$$\n",
    "\n",
    "then \n",
    "\n",
    "$$x = X \\wedge E_0\\, E_0^{-1}$$\n",
    "\n",
    "\n",
    "To implement this in `clifford`  we could create a CGA by instantiating the it directly, like `Cl(3,1)`  for example, and then making the definitions and maps described above relating the various subspaces. Or, we you can use the helper function `conformalize()`. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To demonstrate we will conformalize $G_2$, producing a CGA of $G_{3,1}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'e1': (1^e1), 'e2': (1^e2), 'e12': (1^e12)}"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from numpy import pi,e\n",
    "from clifford import Cl, conformalize\n",
    "\n",
    "G2, blades_g2 = Cl(2)\n",
    "\n",
    "blades_g2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'e1': (1^e1),\n",
       " 'e2': (1^e2),\n",
       " 'e3': (1^e3),\n",
       " 'e4': (1^e4),\n",
       " 'e12': (1^e12),\n",
       " 'e13': (1^e13),\n",
       " 'e14': (1^e14),\n",
       " 'e23': (1^e23),\n",
       " 'e24': (1^e24),\n",
       " 'e34': (1^e34),\n",
       " 'e123': (1^e123),\n",
       " 'e124': (1^e124),\n",
       " 'e134': (1^e134),\n",
       " 'e234': (1^e234),\n",
       " 'e1234': (1^e1234)}"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "G2c, blades_g2c, stuff = conformalize(G2)\n",
    "\n",
    "blades_g2c   #inspect the CGA blades"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$e_{4}$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "print_ga(blades_g2c['e4'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'ep': (1^e3),\n",
       " 'en': (1^e4),\n",
       " 'eo': -(0.5^e3) + (0.5^e4),\n",
       " 'einf': (1^e3) + (1^e4),\n",
       " 'E0': (1.0^e34),\n",
       " 'up': <function clifford.conformalize.<locals>.up(x)>,\n",
       " 'down': <function clifford.conformalize.<locals>.down(x)>,\n",
       " 'homo': <function clifford.conformalize.<locals>.homo(x)>,\n",
       " 'I_ga': (1.0^e12)}"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "stuff"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It contains the following:\n",
    "\n",
    "* `ep` - postive basis vector added\n",
    "* `en` - negative basis vector added\n",
    "* `eo` - zero vecror of null basis (=.5*(en-ep))\n",
    "* `einf` - infinity vector of null basis (=en+ep)\n",
    "* `E0` - minkowski bivector (=einf^eo)\n",
    "* `up()` - function to up-project a vector from GA to CGA\n",
    "* `down()` - function to down-project a vector from CGA to GA\n",
    "* `homo()` - function ot homogenize a CGA vector"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![](http://clifford.readthedocs.io/en/latest/_static/conformal%20space.svg)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "locals().update(blades_g2c)\n",
    "locals().update(stuff)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = e1+e2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$e_{1} + e_{2}$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "print_ga(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$e_{1} + e_{2} + 0.5 e_{3} + 1.5 e_{4}$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "print_ga(up(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$e_{1} + e_{2}$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "print_ga(down(up(x)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "a= 1*e1 + 2*e2\n",
    "b= 3*e1 + 4*e2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$e_{1} + 2 e_{2}$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$$3 e_{1} + 4 e_{2}$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "print_ga(a, b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$0.2 e_{1} + 0.4 e_{2}$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$$0.2 e_{1} + 0.4 e_{2}$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "print_ga(down(ep*up(a)*ep), a.inv())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ - e_{1} - 2 e_{2}$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$$ - e_{1} - 2 e_{2}$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "print_ga(down(E0*up(a)*E0), -a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Dilations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$D_{\\alpha}  =  e^{-\\frac{\\ln{\\alpha}}{2} \\,E_0} $$\n",
    "\n",
    "$$D_{\\alpha} \\, X \\, \\tilde{D_{\\alpha}} $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$0.499646 e_{1} + 0.999292 e_{2}$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$$0.499646 e_{1} + 0.999292 e_{2}$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy import rand,log\n",
    "\n",
    "D = lambda alpha: e**((-log(alpha)/2.)*(E0)) \n",
    "alpha = rand()\n",
    "print_ga(down( D(alpha)*up(a)*~D(alpha)), (alpha*a))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Translations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ V = e ^{\\frac{1}{2} e_{\\infty} a } = 1 + e_{\\infty}a$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$4 e_{1} + 6 e_{2}$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$$4 e_{1} + 6 e_{2}$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "T = lambda x: e**(1/2.*(einf*x)) \n",
    "print_ga(down( T(a)*up(b)*~T(a)), b+a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "from pprint import pprint"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "{'layout': Layout([1, 1, 1, -1], [(), (1,), (2,), (3,), (4,), (1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4), (1, 2, 3), (1, 2, 4), (1, 3, 4), (2, 3, 4), (1, 2, 3, 4)], firstIdx=1, names=['', 'e1', 'e2', 'e3', 'e4', 'e12', 'e13', 'e14', 'e23', 'e24', 'e34', 'e123', 'e124', 'e134', 'e234', 'e1234']),\n",
      " 'value': array([0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0])}\n"
     ]
    }
   ],
   "source": [
    "pprint(vars(einf))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$e_{3}$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$$e_{4}$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$$ - 0.5 e_{3} + 0.5 e_{4}$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "print_ga(ep, en, eo)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Transversions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A transversion is an inversion,  followed by a translation, followed by a inversion. The verser is \n",
    "\n",
    "$$V= e_+ T_a e_+$$\n",
    "\n",
    "which is recognised as the translation bivector reflected in the $e_+$ vector. From the diagram, it is seen that this is equivalent to the bivector in $x\\wedge e_o$,\n",
    "\n",
    "$$ e_+ (1+e_{\\infty}a)e_+ $$\n",
    "\n",
    "$$ e_+^2 + e_+e_{\\infty}a e_+$$\n",
    "$$2 +2e_o a$$\n",
    "\n",
    "the factor of 2 may be dropped, because the conformal vectors are null\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$0.189189 e_{1} + 0.364865 e_{2}$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$$0.189189 e_{1} + 0.364865 e_{2}$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "V = ep * T(a) * ep\n",
    "assert ( V == 1+(eo*a))\n",
    "\n",
    "K = lambda x: 1+(eo*a) \n",
    "\n",
    "B= up(b)\n",
    "print_ga( down(K(a)*B*~K(a)) , 1/(a+1/b) ) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$e_{1} + 2 e_{2}$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$$0.2 e_{1} + 0.4 e_{2}$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "print_ga(a, 1/a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$e_{1}$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$$e_{2}$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
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       "$$e_{12}$$"
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       "$$e_{12}$$"
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   ],
   "source": [
    "print_ga(e1,e2, e1 | e2, e1^e2, e1 * e2)"
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   "execution_count": 25,
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      "text/plain": [
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    {
     "data": {
      "text/latex": [
       "$$ - 2 e_{12}$$"
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   "execution_count": null,
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   "source": []
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  {
   "cell_type": "code",
   "execution_count": 134,
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    "soa = np.array([[0, 0, 1, 1, -2, 0], [0, 0, 2, 1, 1, 0],\n",
    "                [0, 0, 3, 2, 1, 0], [0, 0, 4, 0.5, 0.7, 0]])\n",
    "\n",
    "X, Y, Z, U, V, W = zip(*soa)\n",
    "fig = plt.figure()\n",
    "ax = fig.add_subplot(111, projection='3d')\n",
    "ax.quiver(X, Y, Z, U, V, W)\n",
    "ax.set_xlim([-1, 0.5])\n",
    "ax.set_ylim([-1, 1.5])\n",
    "ax.set_zlim([-1, 8])\n",
    "plt.show()"
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   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$e_{1} + 2 e_{2}$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
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     "metadata": {},
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    }
   ],
   "source": [
    "print_ga(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {},
   "outputs": [],
   "source": [
    "def to_vector(ga, max=-1):\n",
    "    vec = [ ga.value[i] for i, t in enumerate(ga.layout.bladeTupList) if len(t) == 1 ]\n",
    "    max = len(vec) if max == -1 else max\n",
    "    return vec[0:max]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1, 2, 0, 0]"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[ a.value[i] for i, t in enumerate(a.layout.bladeTupList) if len(t) == 1 ]"
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  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
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      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
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    "to_vector(a)"
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  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1, 2]"
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     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
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    "to_vector(a, max=2)"
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   "execution_count": 152,
   "metadata": {},
   "outputs": [],
   "source": [
    "def plot_as_vector(*gas):\n",
    "    fig = plt.figure()\n",
    "    ax = fig.add_subplot(111, projection='3d')\n",
    "    l = len(gas)\n",
    "    # print_ga(*gas)\n",
    "    arr = [to_vector(ga, max=3) for ga in gas]\n",
    "    soa = np.array(arr)\n",
    "    ga_latexes = str_ga(*gas)\n",
    "    for i, v in enumerate(arr):\n",
    "        ax.text(v[0], v[1], v[2], ('$ %s $' % ga_latexes[i]))\n",
    "    # print(soa)\n",
    "    X, Y, Z = np.zeros(l), np.zeros(l), np.zeros(l)\n",
    "    U, V, W = [list(a) for a in zip(*soa)]\n",
    "    # print(X, Y, Z, U, V, W)\n",
    "    ax.quiver(X, Y, Z, U, V, W)\n",
    "    xlim = max(abs(min(U)), abs(max(U)), 2)\n",
    "    ylim = max(abs(min(V)), abs(max(V)), 2)\n",
    "    zlim = max(abs(min(W)), abs(max(W)), 2)\n",
    "    ax.set_xlim([-xlim, xlim])\n",
    "    ax.set_ylim([-ylim, ylim])\n",
    "    ax.set_zlim([-zlim, zlim])\n",
    "\n",
    "#     ax.set_xlim([-1, 0.5])\n",
    "#     ax.set_ylim([-1, 1.5])\n",
    "#     ax.set_zlim([-1, 8])\n",
    "    plt.show()"
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   "cell_type": "code",
   "execution_count": 153,
   "metadata": {},
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   ],
   "source": [
    "plot_as_vector(a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Reflections"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ -mam^{-1}  \\rightarrow MA\\tilde{M}  $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ - 1.78689 e_{1} - 1.34426 e_{2}$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
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     "metadata": {},
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       "$$ - 1.78689 e_{1} - 1.34426 e_{2}$$"
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       "<IPython.core.display.Math object>"
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   ],
   "source": [
    "m = 5*e1 + 6*e2\n",
    "n = 7*e1 + 8*e2\n",
    "\n",
    "\n",
    "print_ga(down(m*up(a)*m), -m*a*m.inv())"
   ]
  },
  {
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   "execution_count": 156,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "['e_{1} + 2 e_{2}', '5 e_{1} + 6 e_{2}', ' - 1.78689 e_{1} - 1.34426 e_{2}']"
      ]
     },
     "execution_count": 156,
     "metadata": {},
     "output_type": "execute_result"
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   "source": [
    "str_ga(a, m, down(m*up(a)*m))"
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  {
   "cell_type": "code",
   "execution_count": 157,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$e_{1} + 2 e_{2}$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
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     "metadata": {},
     "output_type": "display_data"
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    {
     "data": {
      "text/latex": [
       "$$5 e_{1} + 6 e_{2}$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
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     },
     "metadata": {},
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    {
     "data": {
      "text/latex": [
       "$$ - 1.78689 e_{1} - 1.34426 e_{2}$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
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   ],
   "source": [
    "print_ga(a, m, down(m*up(a)*m))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 158,
   "metadata": {},
   "outputs": [
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   ],
   "source": [
    "plot_as_vector(a)\n",
    "plot_as_vector(m)\n",
    "plot_as_vector(down(m*up(a)*m))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Rotations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ mnanm = Ra\\tilde{R} \\rightarrow RA\\tilde{R}  $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 160,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ - 2 e_{1} + 1 e_{2}$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
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   ],
   "source": [
    "R = lambda theta: e**((-.5*theta)*(e12))\n",
    "theta = pi/2\n",
    "print_ga(down( R(theta)*up(a)*~R(theta)))\n",
    "print_ga(R(theta)*a*~R(theta))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 163,
   "metadata": {},
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     "metadata": {},
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   ],
   "source": [
    "plot_as_vector(a, down( R(theta)*up(a)*~R(theta)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 166,
   "metadata": {},
   "outputs": [],
   "source": [
    "from ipywidgets import interact, interactive, fixed, interact_manual\n",
    "import ipywidgets as widgets"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 170,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "f48ca5e5958e4c26a1764503d317c728",
       "version_major": 2,
       "version_minor": 0
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      "text/plain": [
       "interactive(children=(FloatSlider(value=1.5707963267948966, description='theta', max=6.283185307179586), Outpu…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def plot_rotate(origin, theta=pi/2):\n",
    "    R = lambda theta: e**((-.5*theta)*(e12))\n",
    "    rotated = (down( R(theta)*up(origin)*~R(theta)))\n",
    "    plot_as_vector(origin, rotated)\n",
    "\n",
    "interactive_plot = interactive(plot_rotate, origin=fixed(a), theta=(0, 2 * pi, 0.1))\n",
    "output = interactive_plot.children[-1]\n",
    "output.layout.height = '350px'\n",
    "interactive_plot"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As a simple example consider the combination  operations of translation,scaling, and inversion. \n",
    "\n",
    "$$b=-2a+e_0 \\quad \\rightarrow \\quad B= (T_{e_0}E_0 D_2) A \\tilde{ (D_2 E_0 T_{e_0})} $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 172,
   "metadata": {},
   "outputs": [],
   "source": [
    "A = up(a)\n",
    "V = T(e1)*E0*D(2)\n",
    "B = V*A*~V\n",
    "assert(down(B) == (-2*a)+e1 )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 173,
   "metadata": {},
   "outputs": [
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    "plot_as_vector(a, down(B))"
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Transversion"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A transversion may be built from a inversion, translation, and inversion. \n",
    "\n",
    "$$c = (a^{-1}+b)^{-1}$$\n",
    "\n",
    "In conformal GA, this is accomplished by \n",
    "\n",
    "$$C = VA\\tilde{V}$$\n",
    "\n",
    "$$V= e_+ T_b e_+$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 174,
   "metadata": {},
   "outputs": [],
   "source": [
    "A = up(a)\n",
    "V = ep*T(b)*ep\n",
    "C = V*A*~V\n",
    "assert(down(C) ==1/(1/a +b))"
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   "cell_type": "code",
   "execution_count": 175,
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